Hamilton-Jacobi Equation [The Physics Travel Guide]

--- equations:hamilton-jacobi_equation [2018/04/15 11:49]
ida [Abstract]
+++ equations:hamilton-jacobi_equation [2018/04/15 11:50]
ida [Abstract]
@@ Line 51: / Line 51: @@
 where $p_1$ is the momentum of the particle going from $q_0$ to $q_1$, at time $t_1$, and
-\begin{align*}
+$$  \frac{\partial W}{\partial q_0^i}&= -(p_0)_i,\qquad\text{(-momentum at time $t_0$)} $$
- \frac{\partial W}{\partial q_0^i}&= -(p_0)_i,\qquad\text{(-momentum at time $t_0$)}\\
+$$ \frac{\partial W}{\partial t_1}&= -H_1,\qquad\text{(-energy at time $t_1$)} $$
- \frac{\partial W}{\partial t_1}&= -H_1,\qquad\text{(-energy at time $t_1$)}\\
+$$ \frac{\partial W}{\partial t_0}&= H_0,\qquad\text{(+energy at time $t_0$)}
- \frac{\partial W}{\partial t_0}&= H_0,\qquad\text{(+energy at time $t_0$)}
+ $$
-\end{align*}
 Take note that $H_1=H_0$ since energy is conserved.  These last four equations are the Hamilton--Jacobi equations.